View source for Regular dodecahedron

{{Short description|Polyhedron with 12 regular pentagonal faces}}
{{Reg polyhedra db|Reg polyhedron stat table|D}}
[[File:Dodecaedro desarrollo.gif|thumb|Animation of a net of a regular (pentagonal) dodecahedron being folded]]
[[File:Regular dodecahedron.stl|thumb|3D model of a regular dodecahedron]]
[[File:Co20L12 movie.gif|thumb|Crystal structure of '''Co20L12''' dodecahedron reported by Kai Wu, Jonathan Nitschke and co-workers at University of Cambridge in ''Nat. Synth.'' '''2023''', DOI:10.1038/s44160-023-00276-9<sup> <ref name="WuKai2023-15"></ref></sup>]]
A '''regular dodecahedron''' or '''pentagonal dodecahedron''' is a [[Dodecahedron|dodecahedron]] that is [[Regular polyhedron|regular]], which is composed of 12 [[Regular polygon|regular]] pentagonal faces, three meeting at each [[Vertex (geometry)|vertex]]. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 [[Face diagonal|face diagonal]]s, 100 [[Space diagonal|space diagonal]]s).<ref>{{citation|title=Platonic & Archimedean Solids|series=Wooden Books|first=Daud|last=Sutton|publisher=Bloomsbury Publishing USA|year=2002|isbn=9780802713865|page=55|url=https://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55}}.</ref>  It is represented by the [[Schläfli symbol]] {5,3}.

==Dimensions==
If the edge length of a regular dodecahedron is <math>a</math>, the [[Radius|radius]] of a [[Circumscribed sphere|circumscribed sphere]] (one that touches the regular dodecahedron at all vertices) is
:<math>r_u = a\frac{\sqrt{3}}{4} \left(1 + \sqrt{5}\right) \approx 1.401\,258\,538 \cdot a</math>
{{OEIS|A179296}}

and the radius of an inscribed sphere ([[Tangent|tangent]] to each of the regular dodecahedron's faces) is
:<math>r_i = a\frac{1}{2} \sqrt{\frac{5}{2} +\frac{11}{10}\sqrt{5}} \approx 1.113\,516\,364 \cdot a</math>

while the midradius, which touches the middle of each edge, is
:<math>r_m = a\frac{1}{4} \left(3 +\sqrt{5}\right) \approx 1.309\,016\,994 \cdot a</math>

These quantities may also be expressed as
:<math>r_u = a\, \frac{\sqrt{3}}{2} \phi</math>
:<math>r_i = a\, \frac{\phi^2}{2 \sqrt{3-\phi}}</math>
:<math>r_m = a\, \frac{\phi^2}{2}</math>

where ''ϕ'' is the [[Golden ratio|golden ratio]].

Note that, given a regular dodecahedron of edge length one, ''r<sub>u</sub>'' is the radius of a circumscribing sphere about a [[Cube|cube]] of edge length ''ϕ'', and ''r<sub>i</sub>'' is the [[Apothem|apothem]] of a regular pentagon of edge length ''ϕ''.

==Surface area and volume==
The [[Surface area|surface area]] ''A'' and the [[Physics:Volume|volume]] ''V'' of a regular dodecahedron of edge length ''a'' are:
:<math>{\displaystyle A = 3\sqrt{25+10\sqrt{5}} a^2 \approx 20.645\,728\,807a^2}</math>
:<math>V = \frac{1}{4} (15+7\sqrt{5}) a^3 \approx 7.663\,118\,9606a^3</math>

Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio. A dodecahedron with an edge length of one unit has the properties:<ref>{{cite book|last=Livio|first=Mario|title=The Golden Ratio: The Story of Phi, the World's Most Astonishing Number|url=https://books.google.com/books?id=bUARfgWRH14C|orig-year=2002|edition=First trade paperback|year=2003|publisher={{wipe|Random House}}|location=New York City|isbn=0-7679-0816-3|pages=70–1}}</ref>
:<math>{\displaystyle A = \frac{15\phi}{\sqrt{3-\phi}}}</math>
:<math>{\displaystyle V = \frac{5\phi^3}{6-2\phi}}</math>

==Two-dimensional symmetry projections==
The ''regular dodecahedron '' has two high orthogonal projections, centered, on [[Vertex (geometry)|vertices]] and pentagonal faces, correspond to the A<sub>2</sub> and H<sub>2</sub> Coxeter planes. The edge-center projection has two orthogonal lines of reflection.
{|class=wikitable width=400
|+ Orthogonal projections
|-
!Centered by
!Vertex
!Face
!Edge
|-
!Image
|[[File:Dodecahedron A2 projection.svg|160px]]
|[[File:Dodecahedron H3 projection.svg|160px]]
|[[File:Dodecahedron t0 e.png|160px]]
|- align=center
!Projective<br />symmetry
|<nowiki></nowiki>3 = [6]
|<nowiki></nowiki>5 = [10]
|[2]
|}

In perspective projection, viewed on top of a pentagonal face, the regular dodecahedron can be seen as a linear-edged [[Schlegel diagram]], or [[Stereographic projection|stereographic projection]] as a [[Spherical polyhedron|spherical polyhedron]]. These projections are also used in showing the four-dimensional [[120-cell|120-cell]], a regular 4-dimensional polytope, constructed from 120 dodecahedra, [[120-cell#Perspective projections|projecting it down to 3-dimensions]].

{|class="wikitable" width=540
!rowspan=2|Projection
!rowspan=2|Orthogonal projection
!colspan=2|Perspective projection
|-
![[Schlegel diagram]]
![[Stereographic projection]]
|- align=center
!Regular dodecahedron
|[[File:Dodecahedron H3 projection.svg|180px]]
|[[File:Dodecahedron schlegel.svg|180px]]
|[[File:Dodecahedron stereographic projection.png|180px]]
|- align=center
!Dodecaplex<br />([[120-cell|120-cell]])
|[[File:120-cell t0 H3.svg|180px]]
|[[File:Schlegel wireframe 120-cell.png|180px]]
|[[File:Stereographic polytope 120cell faces.png|180px]]
|}

==Spherical tiling==
The regular dodecahedron can also be represented as a spherical tiling.
{|class=wikitable
|- align=center
|[[File:Uniform tiling 532-t0.png|160px]]
|[[File:Dodecahedron stereographic projection.svg|160px]]
|-
![[Orthographic projection]]
![[Stereographic projection]]
|}

==Cartesian coordinates==
[[File:Dodecahedron vertices.png|thumb|240px|
{|
|- valign=top
| colspan=2 |Vertex coordinates:
|- valign=top
|{{Color box|#E46C00|border=darkgray}} || The orange vertices lie at (±1, ±1, ±1) and form a cube (dotted lines).
|- valign=top
|{{Color box|#00FF00|border=darkgray}} || The green vertices lie at (0, ±''ϕ'', ±{{sfrac|1|''ϕ''}}) and form a rectangle on the ''yz''-plane.
|- valign=top
|{{Color box|#0000FF|border=darkgray}} || The blue vertices lie at (±{{sfrac|1|''ϕ''}}, 0, ±''ϕ'') and form a rectangle on the ''xz''-plane.
|- valign=top
|{{Color box|#FF0066|border=darkgray}} || The pink vertices lie at (±''ϕ'', ±{{sfrac|1|''ϕ''}}, 0) and form a rectangle on the ''xy''-plane.
|- valign=top
| colspan=2 | The distance between adjacent vertices is {{sfrac|2|''ϕ''}}, and the distance from the origin to any vertex is {{radic|3}}.<br />{{nowrap|''ϕ'' {{=}} {{sfrac|1 + {{radic|5}}|2}}}} is the golden ratio.
|}
]]
The following Cartesian coordinates define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented:<ref>{{MathWorld|title=Icosahedral group|urlname=IcosahedralGroup}}</ref>
:(±1, ±1, ±1)
:(0, ±''ϕ'', ±{{sfrac|1|''ϕ''}})
:(±{{sfrac|1|''ϕ''}}, 0, ±''ϕ'')
:(±''ϕ'', ±{{sfrac|1|''ϕ''}}, 0)
where {{nowrap|''ϕ'' {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618}} is the [[Golden ratio|golden ratio]]. The edge length is {{nowrap|{{sfrac|2|''ϕ''}} {{=}} {{radic|5}} − 1}}. The circumradius is&nbsp;{{radic|3}}.

===Facet-defining equations===
Similar to the symmetry of the vertex coordinates, the equations of the twelve facets of the regular dodecahedron also display symmetry in their coefficients:
:''x'' ± ''ϕy'' = ±''ϕ''<sup>2</sup>
:''y'' ± ''ϕz'' = ±''ϕ''<sup>2</sup>
:''z'' ± ''ϕx'' = ±''ϕ''<sup>2</sup>

==Properties==
*The [[Dihedral angle|dihedral angle]] of a regular dodecahedron is 2&nbsp;arctan(''ϕ'') or approximately {{val|116.565}}° (where again ''ϕ'' = {{sfrac|1 + {{radic|5}}|2}}, the golden ratio). {{OEIS2C|A137218}} Note that the tangent of the dihedral angle is exactly −2.
*If the original regular dodecahedron has edge length 1, its dual [[Regular icosahedron|icosahedron]] has edge length ''ϕ''.
*If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges. It is the ''roundest'' of the five Platonic solids, enclosing the most volume within the same radius.
*It has 43,380 [[Net (polyhedron)|nets]].
*The map-coloring number of a regular dodecahedron's faces is 4.
*The distance between the vertices on the same face not connected by an edge is ''ϕ'' times the edge length, because the diagonal of a pentagon is ''ϕ'' times its edge length.
*If two edges share a common vertex, then the midpoints of those edges form a 36-72-72 [[Golden triangle (mathematics)|golden triangle]] with the body center.

=== As a configuration ===
This [[Platonic solid#As a configuration|configuration matrix]] represents the dodecahedron. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole dodecahedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element.<ref>{{Cite book | last=Coxeter | first=H.S.M. |  year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book) | chapter=§1.8 Configurations}}</ref><ref>{{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd | isbn= | title-link= | page=117}}</ref>

<math>\begin{bmatrix}\begin{matrix}20 & 3 & 3 \\ 2 & 30 & 2 \\ 5 & 5 & 12  \end{matrix}\end{bmatrix}</math>

Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[Coxeter group]] H<sub>3</sub>, order 120, divided by the order of the subgroup with mirror removal.
{| class=wikitable
!H<sub>3</sub>||{{CDD|node_1|5|node|3|node}}
! ''k''-face||f<sub>k</sub>||f<sub>0</sub> || f<sub>1</sub>||f<sub>2</sub>||[[Vertex figure|''k''-fig]]
!Notes
|- align=center
|A<sub>2</sub> || {{CDD|node_x|2|node|3|node}} || ( ) 
!f<sub>0</sub>
|| 20 || 3 || 3 ||{3} || H<sub>3</sub>/A<sub>2</sub> = 120/6 = 20
|- align=center
|A<sub>1</sub>A<sub>1</sub> ||{{CDD|node_1|2|node_x|2|node}} || { }
!f<sub>1</sub>
||   2 || 30 || 2 || { } || H<sub>3</sub>/A<sub>1</sub>A<sub>1</sub> = 120/4 = 30
|- align=center
|H<sub>2</sub> ||{{CDD|node_1|5|node|2|node_x}} ||[[Pentagon|{5}]]
!f<sub>2</sub>
||   5 ||   5 || 12 || ( ) || H<sub>3</sub>/H<sub>2</sub> = 120/10 = 12
|}

==Geometric relations==
The ''regular dodecahedron'' is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a [[Pentagonal trapezohedron|pentagonal trapezohedron]].

The [[Stellation|stellation]]s of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra.

A [[Rectification (geometry)|rectified]] regular dodecahedron forms an [[Icosidodecahedron|icosidodecahedron]].

The regular dodecahedron has [[Icosahedral symmetry|icosahedral symmetry]] I<sub>h</sub>, [[Coxeter group]] [5,3], order 120, with an abstract group structure of [[Alternating group|''A''<sub>5</sub>]] × [[Cyclic group|''Z''<sub>2</sub>]].

===Relation to the regular icosahedron===
The dodecahedron and icosahedron are [[Dual polyhedron|dual polyhedra]]. A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges.

When a regular dodecahedron is inscribed in a [[Sphere|sphere]], it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%).

A regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...), which ratio is approximately {{val|3.51246117975}}, or in exact terms: {{nowrap|{{sfrac|3|5}}(3''ϕ'' + 1)}} or {{nowrap|(1.8''ϕ'' + 0.6)}}.

===Relation to the nested cube===
A cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions.<ref>http://mathworld.wolfram.com/images/eps-gif/DodecahedronCube_700.gif {{Bare URL image|date=March 2022}}</ref> In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the [[Compound of five cubes|compound of five cubes]].

The ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a regular dodecahedron is 1&nbsp;:&nbsp;''ϕ'', or (''ϕ''&nbsp;−&nbsp;1)&nbsp;:&nbsp;1.

The ratio of a regular dodecahedron's volume to the volume of a cube embedded inside such a regular dodecahedron is 1&nbsp;:&nbsp;{{sfrac|2|2&nbsp;+&nbsp;''ϕ''}}, or {{sfrac|1&nbsp;+&nbsp;''ϕ''|2}}&nbsp;:&nbsp;1, or (5&nbsp;+&nbsp;{{radic|5}})&nbsp;:&nbsp;4.

For example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64&nbsp;+&nbsp;32''ϕ'' (and edge length of 4''ϕ''&nbsp;−&nbsp;4).

Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times&nbsp;''ϕ''.

From these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length ''a'' in terms of the golden mean:
:''V'' = (''aϕ'')<sup>3</sup> · {{sfrac|1|4}}(5&nbsp;+&nbsp;{{radic|5}})
:''V'' = {{sfrac|1|4}}(14''ϕ''&nbsp;+&nbsp;8)''a''<sup>3</sup>

===Relation to the regular tetrahedron===
[[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.]]
As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair).{{Blockquote|text=Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[Compound of five octahedra|compound of five octahedra]], which comes under our definition of stellated icosahedron. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated triacontahedron.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a stella octangula, thus forming a [[Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''chiral''.<ref>{{Cite book | last1=Coxeter | first1=H.S.M. |  last2=du Val | first2=Patrick |  last3=Flather | first3=H.T. | last4=Petrie | author4-link=W:John Flinders Petrie | first4=J.F. | year=1938 | title=The Fifty-Nine Icosahedra | publisher=University of Toronto Studies (Mathematical Series) | volume=6 | page=4 }}</ref>}}

===Relation to the golden rectangle===
[[Golden rectangle]]s of ratio (''ϕ''&nbsp;+&nbsp;1)&nbsp;:&nbsp;1 and ''ϕ''&nbsp;:&nbsp;1 also fit perfectly within a regular dodecahedron.<ref>{{Cite web |last=Knott |first=Ron |date=26 September 2016 |title=The Golden Geometry of Solids or Phi in 3 dimensions |url=https://r-knott.surrey.ac.uk/Fibonacci/phi3DGeom.html#:~:text=and%20vice-versa.-,4.1%20Golden%20sections%20in%20the%20Dodecahedron,%20Icosahedron%20and%20Octahedron,the%20ratio%201%20to%20Phi. |access-date=2022-03-19 |website=Ron Knott's Mathematics Pages}}</ref> In proportion to this golden rectangle, an enclosed cube's edge is ''ϕ'', when the long length of the rectangle is ''ϕ''&nbsp;+&nbsp;1 (or ''ϕ''<sup>2</sup>) and the short length is&nbsp;1 (the edge shared with the regular dodecahedron).

In addition, the center of each face of the regular dodecahedron form three intersecting golden rectangles.<ref>http://www.toshen.com/images/dodecahedronwithgoldrectang.gif {{Bare URL image|date=March 2022}}</ref>

===Relation to the 6-cube and rhombic triacontahedron===
[[File:6demicube-even-dodecahedron.png|240px|thumb|Projection of 6-demicube into regular dodecahedral envelope]]
It can be projected to 3D from the 6-dimensional [[6-demicube|6-demicube]] using the same basis vectors that form the hull of the [[Rhombic triacontahedron#6-cube|rhombic triacontahedron]] from the [[6-cube|6-cube]]. Shown here including the inner 12 vertices, which are not connected by the outer hull edges of 6D norm length {{radic|2}}, form a [[Regular icosahedron|regular icosahedron]].

The 3D projection basis vectors [''u'',''v'',''w''] used are:
:''u'' = (1, ''ϕ'', 0, −1, ''ϕ'', 0)
:''v'' = (''ϕ'', 0, 1, ''ϕ'', 0, −1)
:''w'' = (0, 1, ''ϕ'', 0, −1, ''ϕ'')
{{-}}

==History and uses==
[[File:Roman dodecahedron.jpg|120px|thumb|[[Roman dodecahedron|Roman dodecahedron]]]]
[[File:Rfel vsesmer front.png|thumb|120px|Omnidirectional sound source]]

Regular dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy.

[[Biography:Iamblichus|Iamblichus]] states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons".<ref>[[Biography:Florian Cajori|Florian Cajori]], ''A History of Mathematics'' (1893)</ref> In ''Theaetetus'', a dialogue of Plato, Plato hypothesized that the classical elements were made of the five uniform regular solids; these later became known as the [[Platonic solid|platonic solid]]s. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven". [[Astronomy:Timaeus (dialogue)|Timaeus]] ({{circa|360 BC}}), as a personage of Plato's dialogue, associates the other four platonic solids with the four [[Philosophy:Classical element|classical element]]s, adding that there is a fifth solid pattern which, though commonly associated with the regular dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."<ref>Plato, ''Timaeus'', Jowett translation [line 1317–8]; the Greek word translated as delineation is ''diazographein'', painting in semblance of life.</ref> [[Biography:Aristotle|Aristotle]] also postulated that the heavens were made of a fifth element, which he called [[Philosophy:Aether (classical element)|aithêr]] (''aether'' in Latin, ''ether'' in American English).

[[Biography:Theaetetus (mathematician)|Theaetetus]] gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
[[Biography:Euclid|Euclid]] completely mathematically described the Platonic solids in the [[Euclid's Elements|''Elements'']], the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.

Regular dodecahedra have been used as dice and probably also as divinatory devices. During the Hellenistic era, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain.

In [[Philosophy:20th-century art|20th-century art]], dodecahedra appear in the work of M. C. Escher, such as his lithographs ''Reptiles'' (1943) and ''Gravitation'' (1952). In Salvador Dalí's painting ''The Sacrament of the Last Supper'' (1955), the room is a hollow regular dodecahedron. Gerard Caris based his entire artistic oeuvre on the regular dodecahedron and the pentagon, which is presented as a new art movement coined as Pentagonism.

[[File:Dodecahedron climbing wall.jpg|thumb|120px|left| A climbing wall consisting of three dodecahedral pieces]]

In modern role-playing games, the regular dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice.

[[Company:Immersive Media Company|Immersive Media Company]], a former Canadian [[Digital imaging|digital imaging]] company, made the Dodeca 2360 camera, the world's first 360° full-motion camera which captures high-resolution video from every direction simultaneously at more than 100 million pixels per second or 30 frames per second.{{Advert inline|date=August 2020}} It is based on regular dodecahedron.{{citation needed|date=August 2020}}

The Megaminx twisty puzzle, alongside its larger and smaller order analogues, is in the shape of a regular dodecahedron.

In the children's novel ''The Phantom Tollbooth'', the regular dodecahedron appears as a character in the land of Mathematics. Each of his faces wears a different expression – ''e.g.'' happy, angry, sad – which he swivels to the front as required to match his mood.

===In nature and supramolecules===
[[Image:Braarudosphaera bigelowii.jpg|thumb|right|The fossil record of the coccolithophore ''Braarudosphaera bigelowii'' goes back 100 million years]]
[[Image:Ho-Mg-ZnQuasicrystal.jpg|thumb|right|The faces of a [[Physics:Holmium–magnesium–zinc quasicrystal|Holmium–magnesium–zinc (Ho-Mg-Zn)]] quasicrystal are true regular pentagons]]

The fossil [[Biology:Coccolithophore|coccolithophore]] ''Braarudosphaera bigelowii'' (see figure), a unicellular coastal phytoplanktonic alga, has a calcium carbonate shell with a regular dodecahedral structure about 10 micrometers across.<ref>Hagino, K., Onuma, R., Kawachi, M. and Horiguchi, T. (2013) "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in ''Braarudosphaera bigelowii'' (Prymnesiophyceae)". ''PLoS One'', '''8'''(12): e81749. {{doi|10.1371/journal.pone.0081749}}.</ref>

Some [[Physics:Quasicrystal|quasicrystal]]s and cages have dodecahedral shape (see figure). Some regular crystals such as [[Chemistry:Garnet|garnet]] and [[Chemistry:Diamond|diamond]] are also said to exhibit "dodecahedral" [[Physics:Crystal habit|habit]], but this statement actually refers to the [[Rhombic dodecahedron|rhombic dodecahedron]] shape.<ref name=cryst1>[http://www.khulsey.com/jewelry/crystal_habit.html#h-5. Dodecahedral Crystal Habit] {{webarchive|url=https://web.archive.org/web/20090412012420/http://www.khulsey.com/jewelry/crystal_habit.html|date=12 April 2009}}</ref> <ref name="WuKai2023-15">Kai Wu, Jonathan Nitschke and co-workers "Systematic construction of progressively larger capsules from a fivefold linking pyrrole-based subcomponent", ''Nature Synthesis'', '''2023''', DOI: https://doi.org/10.1038/s44160-023-00276-9.</ref>

==={{Anchor|Shape of the Universe}}Shape of the universe===
Various models have been proposed for the global geometry of the universe. In addition to the primitive geometries, these proposals include the [[Homology sphere#Cosmology|Poincaré dodecahedral space]], a positively curved space consisting of a regular dodecahedron whose opposite faces correspond (with a small twist). This was proposed by Jean-Pierre Luminet and colleagues in 2003,<ref name= "physwebLum03">{{cite web|last1=Dumé|first1=Belle|title=Is The Universe A Dodecahedron?|url=http://physicsworld.com/cws/article/news/2003/oct/08/is-the-universe-a-dodecahedron|website=PhysicsWorld|archive-url=https://web.archive.org/web/20120425223703/http://physicsworld.com/cws/article/news/2003/oct/08/is-the-universe-a-dodecahedron|archive-date=2012-04-25|date=Oct 8, 2003}}</ref><ref name="Nat03">{{cite journal
|last=Luminet
|first=Jean-Pierre
|title=Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
|journal=Nature
|volume=425
|issue=6958
|pages=593–5
|date=2003-10-09
|pmid=14534579
|arxiv=astro-ph/0310253
|doi=10.1038/nature01944
|bibcode=2003Natur.425..593L|s2cid=4380713
}}</ref> and an optimal orientation on the sky for the model was estimated in 2008.<ref name="RBSG08">{{cite journal
|last=Roukema
|first=Boudewijn
|author2=Zbigniew Buliński|author3=Agnieszka Szaniewska|author4=Nicolas E. Gaudin
|title=A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data
|journal=Astronomy and Astrophysics
|volume=482
|pages=747
|year=2008
|arxiv=0801.0006
|doi=10.1051/0004-6361:20078777
|bibcode=2008A&A...482..747L
|issue=3|s2cid=1616362
}}</ref>

In [[Biography:Bertrand Russell|Bertrand Russell]]'s 1954 short story "The Mathematician's Nightmare: The Vision of Professor Squarepunt", the number 5 said: "I am the number of fingers on a hand. I make pentagons and pentagrams. And but for me dodecahedra could not exist; and, as everyone knows, the universe is a dodecahedron. So, but for me, there could be no universe."

==Space filling with cube and bilunabirotunda==
Regular dodecahedra fill space with [[Cube|cube]]s and [[Bilunabirotunda|bilunabirotunda]]s ([[Johnson solid]] 91), in the ratio of 1 to 1 to 3.<ref>{{Cite web|url=http://demonstrations.wolfram.com/DodecahedronAndBilunabirotunda/|title = Dodecahedron and Bilunabirotunda – Wolfram Demonstrations Project}}</ref><ref>http://www.lcv.ne.jp/~hhase/memo/m09_08b.html</ref> The dodecahedra alone make a lattice of edge-to-edge [[Dodecahedron#Pyritohedron|pyritohedra]]. The bilunabirotundas fill the rhombic gaps. Each cube meets six bilunabirotundas in three orientations.
{| class=wikitable width=760
|- valign=top
|[[File:J91.jpg|160px]]<br />Block model
|[[File:Honeycomb of regular dodecahedra-cubes-J91.png|200px]]
|[[File:Dodecahedron lattice.png|200px]]<br />Lattice of dodecahedra
|[[File:Bilunabirotunda augmented cube.png|200px]]<br />6 bilunabirotundas around a cube
|}

==Related polyhedra and tilings==
The regular dodecahedron is topologically related to a series of tilings by [[Vertex figure|vertex figure]] ''n''<sup>3</sup>.

The regular dodecahedron can be transformed by a [[Truncation (geometry)|truncation]] sequence into its [[Dual polyhedron|dual]], the icosahedron:


The regular dodecahedron is a member of a sequence of otherwise non-uniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.''n''). (For ''n''&nbsp;>&nbsp;6, the sequence consists of tilings of the hyperbolic plane.) These face-transitive figures have (''n''32) rotational [[Orbifold notation|symmetry]].

==Vertex arrangement==
The regular dodecahedron shares its [[Vertex arrangement|vertex arrangement]] with four nonconvex uniform polyhedra and three [[Uniform polyhedron compound|uniform polyhedron compound]]s.

Five [[Cube|cube]]s fit within, with their edges as diagonals of the regular dodecahedron's faces, and together these make up the regular polyhedral compound of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a regular dodecahedron.
{| class="wikitable" width=500
| [[File:Great stellated dodecahedron.png|125px]]<br />[[Great stellated dodecahedron]]
| [[File:Small ditrigonal icosidodecahedron.png|125px]]<br />[[Small ditrigonal icosidodecahedron]]
| [[File:Ditrigonal dodecadodecahedron.png|125px]]<br />[[Ditrigonal dodecadodecahedron]]
| [[File:Great ditrigonal icosidodecahedron.png|125px]]<br />[[Great ditrigonal icosidodecahedron]]
|-
| [[File:Compound of five cubes.png|125px]]<br />[[Compound of five cubes]]
| [[File:Compound of five tetrahedra.png|125px]]<br />[[Compound of five tetrahedra]]
| [[File:Compound of ten tetrahedra.png|125px]]<br />[[Compound of ten tetrahedra]]
|}

==Stellations==
The 3 [[Stellation|stellation]]s of the regular dodecahedron are all regular (nonconvex) polyhedra: ([[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]])
{| class="wikitable" width=500
!
!0
!1
!2
!3
|- align=center
|Stellation
| [[File:Dodecahedron.png|125px]]<br />''Regular dodecahedron''
| [[File:Small stellated dodecahedron.png|125px]]<br />[[Small stellated dodecahedron]]
| [[File:Great dodecahedron.png|125px]]<br />[[Great dodecahedron]]
| [[File:Great stellated dodecahedron.png|125px]]<br />[[Great stellated dodecahedron]]
|-
|Facet diagram
|[[File:Zeroth stellation of dodecahedron facets.svg|125px]]
|[[File:First stellation of dodecahedron facets.svg|125px]]<!--using WP but a more consistent alternative exists at commons-->
|[[File:Second stellation of dodecahedron facets.svg|125px]]<!--using WP but a more consistent alternative exists at commons-->
|[[File:Third stellation of dodecahedron facets.svg|125px]]
|}

==Dodecahedral graph==
{{Infobox graph
| name = Regular dodecahedron graph
| image = [[File:Hamiltonian path.svg|160px]]
| image_caption = A Hamiltonian cycle in a dodecahedron.
| namesake =
| vertices = 20
| edges = 30
| radius = 5
| diameter = 5
| girth = 5
| chromatic_number = 3
| chromatic_index =
| fractional_chromatic_index =
| properties = Hamiltonian, [[Regular graph|regular]], [[Symmetric graph|symmetric]], [[Distance-regular graph|distance-regular]], [[Distance-transitive graph|distance-transitive]], [[K-vertex-connected graph|3-vertex-connected]], [[Planar graph|planar graph]]
}}
The [[N-skeleton|skeleton]] of the dodecahedron (the vertices and edges) form a [[Graph (discrete mathematics)|graph]]. It is one of 5 [[Platonic graph]]s, each a skeleton of its [[Platonic solid]].

This graph can also be constructed as the [[Generalized Petersen graph|generalized Petersen graph]] ''G''(10,2) where the vertices of a decagon are connected to those of two pentagons, one pentagon connected to odd vertices of the decagon and the other pentagon connected to the even vertices. Geometrically, this can be visualized as the 10-vertex equatorial belt of the dodecahedron connected to the two 5-vertex polar regions, one on each side.

The high degree of symmetry of the polygon is replicated in the properties of this graph, which is [[Distance-transitive graph|distance-transitive]], [[Distance-regular graph|distance-regular]], and [[Symmetric graph|symmetric]]. The [[Graph automorphism|automorphism group]] has order 120. The vertices can be [[Graph coloring|colored]] with 3 colors, as can the edges, and the diameter is 5.<ref>{{MathWorld|urlname=DodecahedralGraph|title=Dodecahedral Graph}}</ref>

The dodecahedral graph is Hamiltonian – there is a cycle containing all the vertices. Indeed, this name derives from a mathematical game invented in 1857 by [[Biography:William Rowan Hamilton|William Rowan Hamilton]], the [[Icosian game|icosian game]]. The game's object was to find a Hamiltonian cycle along the edges of a dodecahedron.

{| class=wikitable
|+ Orthogonal projection
|[[File:Dodecahedron H3 projection.svg|160px]]
|}
{{-}}

==See also==
*[[120-cell|120-cell]], a regular polychoron (4D polytope whose surface consists of 120 dodecahedral cells)
* {{em|Braarudosphaera bigelowii}} − A dodecahedron shaped [[Biology:Coccolithophore|coccolithophore]] (a [[Biology:Unicellular organism|unicellular]] [[Earth:Phytoplankton|phytoplankton]] [[Biology:Algae|algae]]).
*[[Chemistry:Dodecahedrane|Dodecahedrane]] (molecule)
*[[Pentakis dodecahedron]]
*[[Snub dodecahedron]]
*[[Truncated dodecahedron]]

==References==
<!--See Wikipedia:Footnotes for instructions.-->
{{Reflist|30em}}

==External links==
*{{MathWorld|urlname=RegularDodecahedron|title=Regular Dodecahedron}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3o5x – doe}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1bk9bWiCSjJz6LpNRYDsAu8YDBWnSMrt0ydjpIfF8jmyc682nzINN9xaGayOA9FBx396IIYMhulg2mGXcK0mAk5Rmo8qm9ut0kE1qP&name=Dodecahedron#applet Editable printable net of a dodecahedron with interactive 3D view]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[https://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra] – Models made with Modular Origami
*[http://polyhedra.org/poly/show/3/dodecahedron Dodecahedron] – 3-d model that works in your browser
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
**[[VRML]]#[http://www.georgehart.com/virtual-polyhedra/vrml/dodecahedron.wrl Regular dodecahedron]
*[http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
*[http://www.bodurov.com/VectorVisualizer/?vectors=-0.94/-2.885/-3.975/-1.52/-4.67/-0.94v-3.035/0/-3.975/-4.91/0/-0.94v3.975/-2.885/-0.94/1.52/-4.67/0.94v1.52/-4.67/0.94/-1.52/-4.67/-0.94v0.94/-2.885/3.975/1.52/-4.67/0.94v-3.975/-2.885/0.94/-1.52/-4.67/-0.94v-3.975/-2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-1.52/4.67/-0.94v-2.455/1.785/3.975/-3.975/2.885/0.94v-2.455/-1.785/3.975/-3.975/-2.885/0.94v-1.52/4.67/-0.94/-0.94/2.885/-3.975v4.91/0/0.94/3.975/-2.885/-0.94v3.975/2.885/-0.94/2.455/1.785/-3.975v2.455/-1.785/-3.975/3.975/-2.885/-0.94v1.52/4.67/0.94/-1.52/4.67/-0.94v3.035/0/3.975/0.94/2.885/3.975v0.94/2.885/3.975/-2.455/1.785/3.975v-2.455/1.785/3.975/-2.455/-1.785/3.975v-2.455/-1.785/3.975/0.94/-2.885/3.975v0.94/-2.885/3.975/3.035/0/3.975v2.455/1.785/-3.975/-0.94/2.885/-3.975v-0.94/2.885/-3.975/-3.035/0/-3.975v-3.035/0/-3.975/-0.94/-2.885/-3.975v-0.94/-2.885/-3.975/2.455/-1.785/-3.975v2.455/-1.785/-3.975/2.455/1.785/-3.97v3.035/0/3.975/4.91/0/0.94v4.91/0/0.94/3.975/2.885/-0.94v3.975/2.885/-0.94/1.52/4.67/0.94v1.52/4.67/0.94/0.94/2.885/3.975 Dodecahedron 3D Visualization]
*[http://www.software3d.com/Stella.php Stella: Polyhedron Navigator]: Software used to create some of the images on this page.
*[http://video.fc2.com/content/20141015mMG9QR5R How to make a dodecahedron from a Styrofoam cube]
*[http://www.friesian.com/elements.htm The Greek, Indian, and Chinese Elements – Seven Element Theory]

{{Polyhedra}}
{{Convex polyhedron navigator}}
{{Polytopes}}

[[Category:Goldberg polyhedra]]
[[Category:Planar graphs]]
[[Category:Platonic solids]]

{{Sourceattribution|Regular dodecahedron}}